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Theorem tfinds 4650
Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
tfinds.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
tfinds.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
tfinds.3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
tfinds.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
tfinds.5  |-  ps
tfinds.6  |-  ( y  e.  On  ->  ( ch  ->  th ) )
tfinds.7  |-  ( Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) )
Assertion
Ref Expression
tfinds  |-  ( A  e.  On  ->  ta )
Distinct variable groups:    x, y    x, A    ch, x    ta, x    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    ch( y)    th( x, y)    ta( y)    A( y)

Proof of Theorem tfinds
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tfinds.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
2 tfinds.4 . 2  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
3 dflim3 4638 . . . . 5  |-  ( Lim  x  <->  ( Ord  x  /\  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
43notbii 287 . . . 4  |-  ( -. 
Lim  x  <->  -.  ( Ord  x  /\  -.  (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
5 iman 413 . . . . 5  |-  ( ( Ord  x  ->  (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  <->  -.  ( Ord  x  /\  -.  ( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y ) ) )
6 eloni 4402 . . . . . . 7  |-  ( x  e.  On  ->  Ord  x )
7 pm2.27 35 . . . . . . 7  |-  ( Ord  x  ->  ( ( Ord  x  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  -> 
( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y ) ) )
86, 7syl 15 . . . . . 6  |-  ( x  e.  On  ->  (
( Ord  x  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
9 tfinds.5 . . . . . . . . 9  |-  ps
10 tfinds.1 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
119, 10mpbiri 224 . . . . . . . 8  |-  ( x  =  (/)  ->  ph )
1211a1d 22 . . . . . . 7  |-  ( x  =  (/)  ->  ( A. y  e.  x  ch  ->  ph ) )
13 nfra1 2593 . . . . . . . . 9  |-  F/ y A. y  e.  x  ch
14 nfv 1605 . . . . . . . . 9  |-  F/ y
ph
1513, 14nfim 1769 . . . . . . . 8  |-  F/ y ( A. y  e.  x  ch  ->  ph )
16 vex 2791 . . . . . . . . . . . . 13  |-  y  e. 
_V
1716sucid 4471 . . . . . . . . . . . 12  |-  y  e. 
suc  y
181rspcv 2880 . . . . . . . . . . . 12  |-  ( y  e.  suc  y  -> 
( A. x  e. 
suc  y ph  ->  ch ) )
1917, 18ax-mp 8 . . . . . . . . . . 11  |-  ( A. x  e.  suc  y ph  ->  ch )
20 tfinds.6 . . . . . . . . . . 11  |-  ( y  e.  On  ->  ( ch  ->  th ) )
2119, 20syl5 28 . . . . . . . . . 10  |-  ( y  e.  On  ->  ( A. x  e.  suc  y ph  ->  th )
)
22 raleq 2736 . . . . . . . . . . . 12  |-  ( x  =  suc  y  -> 
( A. z  e.  x  [ z  /  x ] ph  <->  A. z  e.  suc  y [ z  /  x ] ph ) )
23 nfv 1605 . . . . . . . . . . . . . . 15  |-  F/ x ch
2423, 1sbie 1978 . . . . . . . . . . . . . 14  |-  ( [ y  /  x ] ph 
<->  ch )
25 sbequ 2000 . . . . . . . . . . . . . 14  |-  ( y  =  z  ->  ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
2624, 25syl5bbr 250 . . . . . . . . . . . . 13  |-  ( y  =  z  ->  ( ch 
<->  [ z  /  x ] ph ) )
2726cbvralv 2764 . . . . . . . . . . . 12  |-  ( A. y  e.  x  ch  <->  A. z  e.  x  [
z  /  x ] ph )
28 cbvralsv 2775 . . . . . . . . . . . 12  |-  ( A. x  e.  suc  y ph  <->  A. z  e.  suc  y [ z  /  x ] ph )
2922, 27, 283bitr4g 279 . . . . . . . . . . 11  |-  ( x  =  suc  y  -> 
( A. y  e.  x  ch  <->  A. x  e.  suc  y ph )
)
3029imbi1d 308 . . . . . . . . . 10  |-  ( x  =  suc  y  -> 
( ( A. y  e.  x  ch  ->  th )  <->  ( A. x  e.  suc  y ph  ->  th ) ) )
3121, 30syl5ibrcom 213 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  =  suc  y  ->  ( A. y  e.  x  ch  ->  th )
) )
32 tfinds.3 . . . . . . . . . . 11  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
3332biimprd 214 . . . . . . . . . 10  |-  ( x  =  suc  y  -> 
( th  ->  ph )
)
3433a1i 10 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  =  suc  y  ->  ( th  ->  ph )
) )
3531, 34syldd 61 . . . . . . . 8  |-  ( y  e.  On  ->  (
x  =  suc  y  ->  ( A. y  e.  x  ch  ->  ph )
) )
3615, 35rexlimi 2660 . . . . . . 7  |-  ( E. y  e.  On  x  =  suc  y  ->  ( A. y  e.  x  ch  ->  ph ) )
3712, 36jaoi 368 . . . . . 6  |-  ( ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y )  -> 
( A. y  e.  x  ch  ->  ph )
)
388, 37syl6 29 . . . . 5  |-  ( x  e.  On  ->  (
( Ord  x  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  ->  ( A. y  e.  x  ch  ->  ph ) ) )
395, 38syl5bir 209 . . . 4  |-  ( x  e.  On  ->  ( -.  ( Ord  x  /\  -.  ( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y ) )  ->  ( A. y  e.  x  ch  ->  ph ) ) )
404, 39syl5bi 208 . . 3  |-  ( x  e.  On  ->  ( -.  Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) ) )
41 tfinds.7 . . 3  |-  ( Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) )
4240, 41pm2.61d2 152 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  ch  ->  ph ) )
431, 2, 42tfis3 4648 1  |-  ( A  e.  On  ->  ta )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623   [wsb 1629    e. wcel 1684   A.wral 2543   E.wrex 2544   (/)c0 3455   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394
This theorem is referenced by:  tfindsg  4651  tfindes  4653  tfinds3  4655  oa0r  6537  om0r  6538  om1r  6541  oe1m  6543  oeoalem  6594  r1sdom  7446  r1tr  7448  alephon  7696  alephcard  7697  alephordi  7701  rdgprc  24151  tartarmap  25888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398
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